in PROBABILITY
CORRELATION MEASURES
Thomas M. LEWIS
Department of Mathematics Furman University
Greenville, South Carolina, U.S.A.
Geoffrey PRITCHARD Department of Statistics University of Auckland Auckland, New Zealand
submitted July 20, 1999, final versionaccepted October 1, 1999
We would like to extend our gratitude to Joel Zinn and Thomas Shlumprecht for inviting us to attend the Workshop in Linear Analysis and Probability (1998) at Texas A&M University.
AMS subject classification: Primary 60E15
Keywords and phrases: correlation measures, Gaussian correlation inequality
Abstract:
We study a class of Borel probability measures, called correlation measures. Our results are of two types: first, we give explicit constructions of non-trivial correlation measures; second, we examine some of the properties of the set of correlation measures. In particular, we show that this set of measures has a convexity property. Our work is related to the so-called Gaussian correlation conjecture.
1
Introduction
In this article, we study a class of Borel probability measures onR
d, which we call correlation
measures. Our work is related to the so-called Gaussian correlation conjecture; to place our results in context, we will review this important conjecture.
Given x, y ∈ R
d, let (x, y) and kxk denote the canonical inner product and norm on
R
d,
respectively. As is customary, givenA, B⊂R
d andt∈
R, we will writetA={ta:a∈A}and
A+B={a+b:a∈A, b∈B}; the setAis said to besymmetricprovided that−A=Aand convexprovided thattA+ (1−t)A⊂Afor eacht∈[0,1].LetCd denote the set of all closed,
convex, symmetric subsets ofR
d, and letγ
dbe the standard Gaussian measure onR
d,that is,
γd(A) =
1 (2π)d2
Z
A
exp −kxk2/2
dx.
TheGaussian correlation conjecturestates that
γd(A∩B)≥γd(A)γd(B) (1.1)
for each pair of setsA, B ∈ Cd,d≥1.Ford= 1,this conjecture is trivially true, and Pitt [5] has shown that it is true ford= 2.Ford≥3,the conjecture remains unsettled, but a variety of partial results are known. Borell [1] establishes (1.1) for setsA andB in a certain class of (not necessarily convex) sets inR
d, which for d= 2 includes all symmetric, convex sets. The
conjecture can be reformulated as follows: if (X1,· · ·, Xn) is a centered, Gaussian random
vector, then
P
max
1≤i≤n|Xi| ≤1
≥P
max
1≤i≤k|Xi| ≤1
P
max
k+1≤i≤n|Xi| ≤1
(1.2)
for each 1≤k < n.Khatri [4] and ˇSid´ak [7, 8] have shown that (1.2) is true fork= 1.In part, the paper of Das Gupta, Eaton, Olkin, Perlman, Savage, and Sobel [2] generalizes the results of Khatri and ˇSid´ak for elliptically contoured distributions.
The recent paper of Schechtman, Schlumprecht and Zinn [6] sheds new light on the Gaussian correlation conjecture. Their results are of two types: first, they show that the conjecture is true whenever the sets satisfy additional geometric restrictions (additional symmetry, centered ellipsoids); second, they show that the conjecture is true provided that the sets are not too large.
Here is the central question of this article: to what extent is the correlation inequality (1.1) a Gaussian result? In other words, are there any non-trivial probability measures on R
d
satisfying (1.1)? We answer the question in the affirmative. We will call a Borel probability measureλonR
d a correlation measureprovided that
λ(A∩B)≥λ(A)λ(B)
for each pair of setsA, B∈ Cd; we will denote the set of all correlation measures onR
d byMd.
In section 2 we give sufficient conditions for membership inMd and show that Md contains non-trivial elements for each d ≥ 2. In section 3, we examine some properties of correla-tion measures. In particular, we show that non-trivial correlacorrela-tion measures have unbounded support, and that Md has a certain convexity property. Using this convexity property, we construct an element of M2 based on a model introduced by Kesten and Spitzer [3]. Our
results can thus be roughly summarized as:
Measures Correlation property
bounded support no (except in dimension 1) exponential tail (including Gaussian) unknown
heavy tail some examples known
3 demonstrates, the measure of the complement of the ball of radiusr must be positive for eachr≥0.Thus it is natural to ask whether there is a minimal rate with which the measure of the complement of the ball of radius r approaches 0. Perhaps the Gaussian measures lie close to, or on, the “boundary” ofMd, which may account for the difficulty of the Gaussian correlation conjecture.
2
The construction of correlation measures
For d ≥ 2, let B[0,1] denote the closed unit ball of R
d; for r ≥ 0, let B[0, r] = rB[0,1].
Throughout this section,µwill denote a spherically-symmetric, Borel probability measure on R
d. Forr≥0,let
F(r) =µ(B[0, r]).
The main result of this section is Theorem 2.2, which gives sufficient conditions onF for µ
to be a correlation measure; through this result, we produce explicit, nontrivial correlation measures.
The proof of Theorem 2.2 rests on a geometric fact, which we describe presently. LetSd−1
denote the unit sphere of R
d. A subset S of
R
d is called a symmetric slab if there exists a
numberh∈[0,+∞] and a v∈Sd−1such that
S=
x∈R
d :|(v, x)| ≤h
The numberh=h(S) is called thehalf-widthofS; whenh= 0,Sis a hyperplane of dimension
d−1. LetSd denote the set of all symmetric slabs inR
d, and, forA∈ Cd,let
ρ(A) = sup{r≥0 :B[0, r]⊂A}
h(A) = inf{h(S) :S∈ Sd, S⊃A}
It is immediate that ρ(A)≤ h(A); in fact, since A is convex and symmetric, ρ(A) =h(A).
Since A is closed, A ⊃ B[0, ρ(A)]; since Sd−1 is compact, there exists a symmetric slab of
half-widthh(A) containingA. We can summarize these findings as follows:
Lemma 2.1 For each A ∈ Cd, there exists a symmetric slab S of half-width ρ(A) such that
B[0, ρ(A)]⊂A⊂S.
Let σ be uniform surface measure on Sd−1, normalized so that σ(Sd−1) = 1. Since µ is
spherically symmetric, we can representµin polar form: for any Borel subsetAofR
d,
µ(A) = Z ∞
0
σ(t−1A∩Sd−1)dF(t). (2.3)
For 0≤t≤1,let
gd(t) =σ{x∈Sd−1:|x1| ≤t}.
This special function may be expressed as
gd(t) =Kd
Z t
0
where
A simpler form of this result can be obtained by strengthening the conditions onF. LetL2= 1
and, ford≥3,letLd=Kd.With this convention,
gd(t)≤Ldt (2.8)
ford≥2 andt∈[0,1].
Corollary 2.3 IfF is concave and
F(b) +Ldb
1 + 1
F(b) Z ∞
b
t−1dF(t)≤1 (2.9)
for eachb∈(0,∞),then µ∈ Md.
Proof We will show that the conditions of Theorem 2.2 are satisfied. SinceF is concave,
F(a)
a ≥ F(b)
b (2.10)
for 0< a≤b.SinceF is ultimately positive, this shows thatF(a)>0 for a >0.
Let 0< a≤b <∞.Then
F(b) + Z ∞
b
gd
b
t
+ 1
F(a)gd a
t
dF(t)
≤F(b) +Ld
b+ a
F(a) Z ∞
b
t−1dF(t) (by (2.8))
≤F(b) +Ldb
1 + 1
F(b) Z ∞
b
t−1dF(t), (by (2.10))
which shows that (2.9) implies (2.5).
Our next result uses Corollary 2.3 to demonstrate the existence of non-trivial correlation measures in each dimensiond≥2.
Theorem 2.4 For each L ≥ 1, there exists a differentiable, concave, increasing function
F : [0,∞)→[0,1]satisfying
F(r) +Lr
1 + 1
F(r) Z ∞
r
F′(t)
t dt≤1 (2.11)
for eachr∈(0,∞). Proof Let
F(r) = (1
2r
1/4L, forr≤1;
This makesF differentiable, concave, and increasing on [0,∞). Forr≥1, the left-hand side
inequality (2.11) becomes an equality. This function F is thus the best possible solution to (2.11) in that sense.
3
Some properties of correlation measures
Letµdenote a Borel probability measure onR
d.As is customary, let thesupport ofµ(denoted
by supp(µ)) be the intersection of the closed subsets ofR
d having full measure.
Theorem 3.1 If µhas compact support and dim (supp(µ))>1,thenµ /∈ Md.
In other words, unless a correlation measure is supported on a one-dimensional subspace, it must have unbounded support.
Proof Let x0 ∈ supp(µ) have maximal distance from 0. Without loss of generality we may
Observe thatAǫ∪Bǫ⊃B[0,1]⊃supp(µ); thus,µ(Acǫ∩Bcǫ) = 0. Since dim (supp(µ))>1,we
can chooseǫ >0 such thatµ(Ac
ǫ∩Bǫ) =µ(Acǫ)>0.Sincee1∈Bǫc, µ(Aǫ∩Bǫc) =µ(Bǫc)>0.
Finally,
µ(Aǫ∩Bǫ)−µ(Aǫ)µ(Bǫ)
=µ(Aǫ∩Bǫ)µ(Acǫ∩Bǫc)−µ(Aǫ∩Bǫc)µ(Acǫ∩Bǫ)<0,
which shows thatµ /∈ Md.
Our next result shows that Md remains closed under certain convex combinations. Let µ
andλbe Borel probability measures onR
d. We will say thatµdominates λ(writtenµ≻λ)
provided thatµ(A)≥λ(A) for eachA∈ Cd.
Theorem 3.2 Let µ, λ ∈ Md with µ ≻ λ, and let a, b be nonnegative real numbers with
a+b= 1.Then aµ+bλ∈ Md.
Proof Letm=aµ+bλ, and letA, B∈ Cd.Then
m(A)m(B) =a2µ(A)µ(B) +abµ(A)λ(B) +abµ(B)λ(A) +b2λ(A)λ(B).
Sincea+b= 1 andµandλare correlation measures,
m(A∩B) = (a+b)m(A∩B)
=a2µ(A∩B) +abµ(A∩B) +abλ(A∩B) +b2λ(A∩B)
≥a2µ(A)µ(B) +abµ(A)µ(B) +abλ(A)λ(B) +b2λ(A)λ(B).
Recalling thatµ≻λ,we have
m(A∩B)−m(A)m(B)
≥ab(µ(A)µ(B) +λ(A)λ(B)−µ(A)λ(B)−µ(B)λ(A)) =ab(µ(A)−λ(A)) (µ(B)−λ(B))≥0,
which shows thatm∈ Md,completing our proof.
In general, a linear combination of correlation measures need not be a correlation measure. For example, letµandλbe the centered Gaussian measures onR
2 with covariance matrices
Qµ=
1 0 0 2
and Qλ=
2 0 0 1
,
respectively. By the theorem of Pitt [5], µ and λ are correlation measures; however, the measurem= (µ+λ)/2 is not a correlation measure. To see this, let
A={(x1, x2)∈R
2 :|x 1| ≤1}
B ={(x1, x2)∈R
2 :|x 2| ≤1}.
Then, by a calculation as in the proof of Theorem 3.2, m(A∩B)−m(A)m(B) <0, which shows thatm /∈ M2.
Corollary 3.3 Let {µi : 1 ≤ i ≤ n} ⊂ Md with µ1 ≻ µ2 ≻ · · · ≻ µn−1 ≻ µn, and let {ai: 1≤i≤n}be a set of nonnegative real numbers withPni=1ai= 1.ThenPni=1aiµi∈ Md.
Dominating measures can be constructed through scaling. Given µ ∈ Md and s > 0, let
µs(A) = µ(sA) for each Borel subset of R
d. If r ≥ s, then rA ⊃ sA for each A ∈ Cd;
thus, µr ≻ µs. We will use this notion of domination through scaling in conjunction with
Corollary 3.3 to construct elements ofM2.
Let{Sn :n≥0}(S0= 0) be simple random walk onZ, and let{Y(k) :k∈Z}be a sequence of independent and identically distributed, two-dimensional, standard Gaussian random vectors. We will assume that the random walk and the Gaussian vectors are defined on a common probability space and generate independent independentσ-algebras. Forn≥0,let
Zn= n
X
k=0
Y(Sk).
The process {Zn :n≥0}, calledrandom walk in random scenery, was introduced by Kesten
and Spitzer [3], who investigated its weak limits.
Theorem 3.4 For each n≥0, the law ofZn is an element ofM2.
Proof Forn≥0,letζn denote the law ofZn.Forj∈Zandn≥0, let
ℓjn= n
X
k=0
I(Sk =j)
and observe thatZn=Pj∈Zℓ
j
nY(j). Forn≥0,let
Vn =
X
j∈Z
ℓjn
2
.
The process{Vn : n≥0} is called the self-intersection local time of the random walk.
Con-ditional on theσ-field generated by the random walk,Zn is a Gaussian random vector with
covariance matrixVn times the identity matrix. Thus, for each Borel setA∈R
2,
ζn(A) =
∞
X
k=0
P(Zn∈A|Vn=k)P(Vn =k)
=
∞
X
k=0
γ2(k−1/2A)P(Vn =k).
By the theorem of Pitt [5], the measures{γ2(k−1/2·) :k ≥ 1} are in M2, and, by scaling,
the measures can be ordered by domination; thus, by Corollary 3.3,ζn is inM2, as was to be
shown.
References
[1] Christer Borell, A Gaussian correlation inequality for certain bodies in R
n, Math. Ann.
[2] S. Das Gupta, M. L. Eaton, I. Olkin, M. Perlman, L. J. Savage, and M. Sobel, Inequal-ities on the probability content of convex regions for elliptically contoured distributions, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 241–265. Univ. California Press, Berkeley, Calif., 1972.
[3] H. Kesten and F. Spitzer,A limit theorem related to a new class of self-similar processes, Z. Wahrsch. Verw. Gebiete50(1979), no. 1, 5–25.
[4] C. G. Khatri, On certain inequalities for normal distributions and their applications to simultaneous confidence bounds, Ann. Math. Statist.38(1967), 1853–1867.
[5] Loren D. Pitt,A Gaussian correlation inequality for symmetric convex sets, Ann. Probab. 5(1977), no. 3, 470–474.
[6] G. Schechtman, Th. Schlumprecht, and J. Zinn,On the Gaussian measure of the intersec-tion, Ann. Probab.26(1998), no. 1, 346–357.
[7] Zbynˇek ˇSid´ak,Rectangular confidence regions for the means of multivariate normal distri-butions, J. Amer. Statist. Assoc.62(1967), 626–633.
